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# Vector potential

In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field.

Formally, given a vector field v, a vector potential is a vector field A such that

\( \mathbf{v} = \nabla \times \mathbf{A}. \)

If a vector field v admits a vector potential A, then from the equality

\( \nabla \cdot (\nabla \times \mathbf{A}) = 0 \)

(divergence of the curl is zero) one obtains

\( \nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0, \)

which implies that v must be a solenoidal vector field.

An interesting question is then if any solenoidal vector field admits a vector potential. The answer is yes, if the vector field satisfies certain conditions.[citation needed]

Theorem

Let

\( \mathbf{v} : \mathbb R^3 \to \mathbb R^3 \)

be a solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as ||x||→∞. Define

\( \mathbf{A} (\mathbf{x}) = \frac{1}{4 \pi} \nabla \times \int_{\mathbb R^3} \frac{ \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d\mathbf{y}. \)

Then, A is a vector potential for v, that is,

\nabla \times \mathbf{A} =\mathbf{v}. \)

A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field.

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is

\( \mathbf{A} + \nabla m \)

where m is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.

This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge.

See also

Fundamental theorem of vector analysis

Magnetic potential

Solenoid

References

Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.

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